The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow
This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca [rightward arrow] 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.
The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [double less-than sign] 10−6. For Qc [double less-than sign] Q [double less-than sign] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [double less-than sign] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.(Published Online April 26 2006)
(Received January 22 1993)
(Revised September 26 1994)
p1 Present address: The Levich Institute, The City College of CUNY, New York, NY 10031, USA